Calc 2 final

Diverge: does not approach a real number (inf,-inf)

converge: approaches a real number (0,1,2)

Type of diverging sequences:

1: oscillation: (0,7,0,7,0,7,0)

2: sequence approaches (inf,-inf)

Value of series = limit of its sequence of partial sums

constant multiple rule

Σcan=CΣan Simplify by factoring out constant

sum rule for series

Σan+bn=Σan+Σbn

Testing for convergence and divergence

nth term test: If the limit of a sequence does not equal 0, the series is divergent. If the limit equals 0, it doesn’t mean it is convergent.

Ratio test

a(n+1)/(an)+1. Multiply by inverse

Area

∫top-bottomdx

To find x bounds

set equations equal to eachother

washers rotating over x axis

y=x

washers rotating over y axis

x=y

Shells rotating over x axis

x=y

shells rotating over y axis

y=x

washer formula:

∫pi(outer²- inner²)dx

Shells formula

∫2pix(outer-inner)dx

moment of Y

∫x(x1-x2)dx

moment of X

½ ∫x1²-x2²dc

when rotating around a line

subtract

arc length

sqrt(1+f’x)dx

conical tank

density∫pi(rx/h)²(h-x)dx

Vectors

dot product:

ab/sqrt(a) x sqrt(b)

finding angle between vectors

theta= cos^-1(dot product)

Area of a parallelogram

||v|| ||w|| sin theta

How to evaluate integrals

integrate by part, use usub, add solutions together.

taylor sequence

f^n’(a)/n! (x-a)^n

maclaurin sequence

(f^n(0)x)’/n!